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I agree with Donu. Indeed, I think even the much weaker question of whether a mod-p representation of the fundamental group of the base on Sp(2g,Z/pZ) occurs as a monodromy representation might typically have a negative answer. Given such a representation rho, you get a fibration X_rho -> P^1, whose fibers are isomorphic to the moduli space of abelian g-folds with full p-level structure; this will be general type for p large. Any abelian g-fold A/C(t) with monodromy rho corresponds to a section from P^1 back to X_rho, and I don't see why there would be such a section in general.